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58
МАТЕМАТИКА
Approximation of Continuous 2n-Periodic Piecewise Smooth Functions by Discrete Fourier Sums
G. G. Akniyev
Gasan G. Akniyev, https://orcid.org/0000-0001-8533-4277, Dagestan Scientific Center RAS, 45 M. Gadzhieva St., 367025 Makhachkala, Russia, hasan.akniyev@gmail.com
Let N be a natural number greater than 1. Select N uniformly distributed points tk = 2nk/N + u (0 < k < N - 1), and denote by LnN(f) = LnN(f,x) (1 < n < N/2) the trigonometric polynomial of order n possessing the least quadratic deviation from f with respect to the system {tk}N—. Select m + 1 points —n = a0 < ai < ... < am-i < am = n, where m ^ 2, and denote O = {a,}m=0. Denote by C^ a class of 2n-periodic continuous functions f, where f is r-times differentiable on each segment A, = [a,, ai+i] and f(r) is absolutely continuous on A,. In the present article we consider the problem of approximation of functions f e C^ by the polynomials Ln,N(f,x). We show that instead of the estimate |f(x) — Ln,N(f,x)| < clnn/n, which follows from the well-known Lebesgue inequality, we found an exact order estimate |f(x) — Ln,N(f,x)| < c/n (x e R) which is uniform with respect to n (1 < n < N/2). Moreover, we found a local estimate |f (x) — Ln,N(f, x)| < c(e)/n2 (|x — a,| ^ e) which is also uniform with respect to n (1 < n < N/2). The proofs of these estimations are based on comparing of approximating properties of discrete and continuous finite Fourier series.
Keywords: function approximation, trigonometric polynomials, Fourier series.
Received: 22.05.2018 / Accepted: 28.11.2018
Published online: 28.02.2019
DOI: https://doi.org/10.18500/1816-9791 -2019-19-1 -4-15
INTRODUCTION
We begin by establishing some notations. Let O be a set of m +1 points {a,}m=0 (m > 2) such that —n = ao < < ai < ... < am-i < am = n. We denote by the class of 2n-periodic functions with r absolutely continuous derivatives on each interval (a, ,ai+i) and by CO the subclass of
all continuous functions in (here we say that a function f is absolutely continuous on an interval (a, b) if the function f is absolutely continuous on the segment [a, b], where f (x) = f (x) for x £ (a, b), f (a) = f (a + 0), and f(b) = f (b - 0)).
We denote by Ln,N(f, x) a trigonometric polynomial of order n possessing the least quadratic deviation from the function f at the points {tjj^-1, where tj = u + 2nj/N, n ^ N/2, N ^ 2, and u £ R. In other words, Ln,N(f, x) provides the minimum for the sum ij-1 |f(tj) — Tn(tj)|2 on the set of all trigonometric polynomials of order at most n. To read more about function approximation by trigonometric polynomials see [1-10].
Also, in this paper we denote by c or c(b1, b2,..., bk) some positive constants, which depend only on specified arguments (if any) and may vary from line to line, and by Sn(f, x) the n-th partial sum of the Fourier series of the function f. We also note that it is easy to show that the Fourier series of any f £ converges uniformly on R and the following representation is possible:
to
f (x) = 70 + ^^ (ak cos kx + bk sin kx), (1)
2 k=i
1 r 1 r
ak = -/ f (t)cos ktdt, bk = -/ f (t) sin ktdt. (2)
n J-n n J-n
where
11
— f (t) cos ktdt, bk = — n J-n n
The goal of this work is to estimate the value |f(x) — Ln,N(f, x)| for f £ . Note that the special case of this problem is considered in [11], where the value |f(x) — Ln,N(f,x)| is estimated for a 2n-periodic function f(x) = |x| (x £ [—n,n]). In this work, we generalize the results from [11] for any function f £ , as stated in the following theorem:
Theorem 1. For f £ the following inequalities hold:
|f(x) — Ln,N(f,x)| < f, x £ R, (3)
n
|f (x) — Ln,N(f, x)| < , x £ |x — a| ^ (4)
The order of these estimates cannot be improved. To prove this theorem we use a lemma from [12]:
Lemma 1 (Sharapudinov, [12]). If the Fourier series of f converges at the points tk = u + 2kn/N, then the representation
Ln,N (f, x) = Sn(f, x) + Rn,N (f, x), (5)
where
2 to }
Rn,N(f, x) = - Y^ Dn(x — t) cos^N(u — t)f (t)dt, (6)
holds true, where 2n < N and Dn (x) is the Dirichlet kernel:
2
Dn (x) = - + ^>s kx. (7)
k=1
This lemma considers only the case 2n < N .If 2n = N (when N is even) we can write (see [12])
Ln,2n (f,x) = Ln-l,2n (f,x) + (f )C0S n(x - u),
where
2n-1
ai2n) (/ ) = 2^E/ (tk )cos "(tk — «)•
k=0
To prove the inequalities (3) and (4) from Theorem 1 we use the formulas |/(x) — Ln,N(/, x)| ^ |/(x) — Sn(/,x)| + K,N(/, x)| , n < N/2,
(2n)
(8)
(9)
(10)
|/(x) — Ln,N(f,x)| < |/(x) — Sn-i(/,x)| + |Rn-i,N(/,x)| + |an (/)| , n = N/2, (11)
which immediately follow from (5) and (8).
The estimates for the values |/(x) — Sn(/,x)|, |Rn,N(/,x)|, and |an2n)(/)| are found in the following sections.
1 . THE ESTIMATE FOR |/(x) — Sn(/, x)|
To estimate the value |/(x) — Sn(/, x)| we need the following lemma.
Lemma 2. For / e C^ the following inequality holds:
f (t)hp (k(t + a))dt
<
cf) k2 '
where k G N, a G R, and
hp (x) =
cos x, p = 0,
sin x, p = 1.
Proof. Performing integration by parts two times we have
(12)
f(t)hp(k(t + a))dt = Ц^- I f (t)hi-p(k(t + a))dt
1 k2
—n m—1
Y. f (« - 0) - f' (« + 0)) hp(k(a, + a)) - f (t)hp(k(t + a))dt i=0 _
From this we can get the estimate
f (t)hp (k(t + a))dt
1
< k12
m—1
£|f' (« - 0) - f (a, + 0) + f (t)
i=0
dt
<
c(f) k2 ■
□
П
П
Lemma 3. For f G C2 the following inequalities hold:
|f (x) - Sn(f,x)| < f, x G R, (13)
n
|f (x) - Sn(f, x)| < fr, |x - O« | > £. (14)
Proof. Here we prove only (13) because the proof for inequality (14) can be found in [13, Theorem 2.1]. Using (1) and (2) we can write
to
f (x) — Sn (f, x) = ^^ (ak cos kx + sin kx).
k=n+1
Applying Lemma 2 to (2) we get |ak| ^ c(f)/k2 and |bk| ^ c(f)/k2, which gives us
to
|f(x) - S„(f,x)| ^ £ (|Ok| + |) < c(f)/n.
k=n+1
□
2 . THE ESTIMATE FOR |Rn,N(f, x)
From (6) and (7) follows (f,x) = Ri(f,x) + R^n(f,x), where
1 to }
(f, x) = f (t) cos mN(u - t)dt,
2 to n "
9 _ _ _ _ /*
R22(f, x) = - ^ f (t) cos k(x - t) cosmN(u - t)dt. (15)
^ ^=1 k=1-n
Obviously, |Rn,N(f,x)| < |Rn,N(f,x)| + |R2,n(f,x)|. The values |Rn,n(f,x)| and |Rn,N(f, x) | are estimated later in this section, but first we prove three auxiliary lemmas.
Lemma 4. For f £ C0'1 the following holds:
f (t)hp(k (t — x))hq (MN (t — u))dt =
—n m—1
( J2 E (f (Oi - 0) - f (Oi + 0)) hp(k(Oi - x))h1—q(mN(Oi - u))-
(mN)2 - k2 i=0
n
- /äff2 / f'(t)hp(k(t - x))h1—q(MN(t - u))dt+ («N) - k2 J
(mN r - '2 ' " P
—n
m—1
(_i)1+pk m—11
+ / ( a 12 , 2E (f (Oi - 0) - f (Oi + 0)) h1— p(k(Oi - x))hq(MN(Oi - u))-
(mn) - k21=0
n
(MN1)2-kkiJ f (t)h1—p(k(t - x))hq(mN(t - «))&. (16)
—n
Изв. Сарат. ун-та. Нов. сер. Сер. Математика. Механика. Информатика. 2019. Т. 19, вып. 1 Proof. Perform integration by parts:
f (t)hp(k(t - X))hq(MN(t - U))dt =
—n
m—1
(_1)q m—1
(f («i - 0) - f («, + 0)) hp(k(a, - x))h1—q(mN(a, - u))-
MN
i=0
- TNT / f' (t)hp (k(t - x))h1—q (mN (t - u))dt+
— П
7Г
+ ( m^V/(t)hi-p(k(t — x))hi-q(mN(t — u))dt. (17)
—n
Repeat integration by parts for the last integral in (17):
I /(t)hp(k(t — x))hq(mN(t — u))dt =
— П
m—1
(_1)q m—1
^ E (f (a, - 0) - f (a, + 0)) hp(k(a, - x))h1—q(mN(a, - u))-
i=0
(-1)q , ,
( ) ' f (t)hp(k(t - x))h1—q(mN(t - u))dt+
MN
— П
m—1
(_1)1+p k m—1
\ ;П2 E (f (a, - 0) - f (a, + 0)) h1— p(k(a, - x))hq(mN(a, - u))-(MN)
,=0
П
-^тНтг? f f'(t)h1—p(k(t - x))hq(mN(t - u))dt+ (mN ) J
(MN)
k2 (mn )
— П
f (t)hp(k(t - x))hq(mN(t - u))dt.
2p — П
By moving the last integral from the right side to the left and dividing both sides by
2 _
^NNT2 we get (16). □
Corollary 1. If f G CQ, then f (a, - 0) - f (a, + 0) = 0, so we can write (16) as
П П
J f (t)hp(k(t - x))hq(MN(t - u))dt = ¿ff -N J' f' (t)hp(k(t - x))h1—q(mN(t - u))dt-
— П —П
(-1)1+p k
- - ■■■"Я—p
J f (t)h1—p(k(t - x))hq(mN(t - u))dt.
(mN)2 - k2
— П
7Г
Lemma 5. The following estimate holds:
1
(kx)
k=1
<
Proof. The proof is obvious and follows from well-known formulas
£ sin(fex)=sin (:;n (f;, ± cos(kx)=cos (^xjsin (nx)
k=1
sin (2)
□
Lemma 6. Let a1, a2,..., an be a monotonous sequence (either increasing or decreasing) of n positive numbers. The following holds:
hp (kx)
k=1
<
2an + a
| sin x
Proof. After performing Abel transformation (summation by parts) we have:
n—1
^ akhp(kx) = an E hP(jx) - S (ak+1 - ak) E hP(jx)-
k=1
=1
k=1
=1
Using Lemma 5 and the fact that ^n=1 |ak+1 — ak| = En=1 ak+1 — ak| we can write
hp (kx)
k=1
< a
n—1
k=1
<
E hp(jx) j=1
' n—1
^«k+1 - ak
an +
+ y] |ak+1 - ak|
2an + a
hp(jx)
j=1
<
k=1
<
'sin x
□
Lemma 7. The following inequality holds: R N(f, x)| ^ c(f)/N2. Proof. Using Lemma 2 we have
Rn(f,x)| < ^
^=1
f (t) cos mN(u - t)dt
< f v 1 < f
< N2 ^ M2 < N2
U = 1
Lemma 8. The following estimates hold:
K,n(f,x)| < ,
x e R,
R2(f,x)| <
f)
N2
, |x - O; | ^ e.
□
(18) (19)
2
1
2
n
Proof. Rewrite (15) using (12):
^ Ж n П
R(f,x) = -££ f(t)ho(k(t - x))ho(MN(t - u))dt.
П
"=1 fc=1_
Using Corollary 1 we rewrite the above formula as follows:
СЛ Ж 1 n -1 /»
R2n,N(f,x) = E " E-f (t)ho(k(t - x))h1 (mN(t - u))dt+
"=1 M k=1 1 - (JN) —П
2 1 k 1 /* +r^E^E N—/' (t)hi (k(t — x))ho (mn (t — =
m k=iN1 — (¿N) ——n
= r^N (/,x) + Rjv (/,x).
For brevity we only consider here estimation of |R22,',n(/, x) | because IR'N(/, x) | can be estimated in almost the same way. Obviously, /' e C^'1, so we can apply Lemma 4 to R" (/, x):
- ж 1 n 1 m—1
(f x) = ^Е зЕ ( - ^E (f' (a® - 0) - f' (a + °))
k=1 (1 - (») '=°
xho(k(a, - x))ho(mN(a, - u)
x
2 1 1 С ^ E E T-^ f''(t)ho(k(t - x))ho(MN(i - u))dt+
k=1 (! - O) )
—2 A 1 -A k m—1
+^ E M3 E T : V E f (a, - 0) - f (a, + 0)) x
k=1 (1 - O)) ,:=o
xh1 (k(a, - x))h1 (mN(a, - u)) +
2 ж 1 n k П
E 7! E T-^ / f''(t)h1 (k(t - x))h1(MN(t - u))dt
nN( l , -2\
" k=111 - O))
= r2,N (f,x) + RtN (f,x) + r2,N (f,x) + r2;,N4 (f,x).
Begin with R2'1v (f,x). Applying Lemma 4 we get
q ж n m—1
R^N (f, x) = ^E ^E t / -^E (f "(a. - 0) - /''(a, + 0)) x
k=1 (1 - O)) !=°
xho(k(a, - x))h1 (mN(a, - u)) + 10 Научный отдел
П
G. G. Akniyev. Approximation of Continuous -Periodic Piecewise Smooth Functions
—2 ^ ^_1
nN3 ^ -3 f^
¿ = 1 ^ k = 1
f''' (t)ho (k(t — x))h1 (-N (t — u))dt+
1 - ( A i i -n
¿N
k
m-1
_2 to i n
+ nN4 ^ J4 k=1 ( u\ 2 . . 0
¿=1 r k=1 M _ / \ 1 i=0
1 Un
sE (f''(ai — 0) — f'' (a, + 0)) x
xh1 (k(a, — x))h0(-N (a, — u))+
2 to i n ttTV4
k
nN4 ^ -4 ^ / / x2\
k=1 (i — (¿v))
f''' (t)h1(k (t — x))ho (-N (t — u))dt.
From this we can get the estimate
to n
c1
m- 1
KN2(f,x)| <
N3 ^ J3 ^ / , ,2 ¿= - k=1 11 — ^
^|f'' (a, — 0) — f'' (a, + 0)
i=0
+
C to 1 n
+ N3 ^ -3 ^ ¿=1 ^ k=1
C to 1 n
+ N3 ^ J4^ ¿=1 ^ k=1
k/N
1 — '¿N
m- 1
fm (t)
dt+
1 — ' ¿N
i=0
T^Ef'' (a' — 0) —
+
TO 1 n
3 E-^E
k/N
N ¿= - " (1 — (*)=)
f'" (t)
dt <
Cf) N2 '
In the same way we can get R^N4(f, x) | ^ c(f )/N2. Now we consider R^N1 (f,x and | Rn,vv3(f, x) |. We will estimate here only R^v1 (f, x) | because the other one can be estimated in the similar way. After a simple transformation we have
2 to 1 m-1
R^'N1 (f, x) = -^E " E f' (a, — 0) — f' (a, + 0)) M/'N (a, — u)) E
n ¿=1 - ,=0
k=1
hp (k(a, — x))
'1 — i ¿v
From this we have the uniform estimate for x R:
to m-1
|
R'N1 (f,x)| < ^E -2 E |f' (a, — 0) — f' (a, +
¿=1 - ,=0
E
k=1
h0 (k(a, — x))
1 — ' ¿N
<
nc(f) N2
n
3
n
3
1
n
1
n
3
2
Maтeмaтnкa
11
Using Lemma 6 and assuming = --1—^ we have
E
k=1
hp(k (a« — x))
<
sin
i-( TN )
2
V 1 — W
T7 +
1
1—
<
I sin
Now we can write
KN1 (f,x)| <
¿=1 ^ i=0
In the similar way we can get
nc(f)
c ^ 1 m-1 If (a« — 0) — f (a« + 0)| < c(f,e)
I sin ^
N2
|x — a«| ^ e.
c(f,e)
(f x)I < ^, x € R and (f,x)I < ^Z, |x — a«| > e.
Finally, for R.'N (f, x) we can write
N2
(f x)| < EKN(f, x) I < if, x € R, IR22',N(f,x)| < f-, |x — a«| ^ e
«=1 N N
Using the same approach we can show that the value IR'jv(f, x) | has the same estimate as IR'iv(f, x)|, which leads us to (18) and (19). □
From the previous lemmas and inequality (f,x)| ^ R(f,x)| + (f,x follow estimates for |Rn>N(f, x) |:
|Rn,N(f,x)| < nf, x € R, |R„,n(f, x)| < , |x — a«| > e.
(20) (21)
3 . THE ESTIMATE FOR
ai2n) (f)
Lemma 9. For the value ai2n) (f) where f
ai2n) (f) < c(f )/N2 holds.
Proof. For each f € Cg the sum S = £k=—1 (ffe) — f(tfe+1)) = 0. We can
represent the above sum as S = S1 + S2, where S1 = ^П=0 (f (t2k) — f (t2k+1)) and
S2 = £n-01 (f (W) - f (t2k+2)). We can see that Si = -S2 and |Si| = |Si - S21 /2. From the above formulas we can write the equation for S1 — S2:
n=1
n=1
S1 — S. = £ (f (t2k) — 2f (i.fc+1) + f (t.k+2)) = £ Д2f (t2k) .
(22)
k=0
k=0
Denote by G the set of numbers Um=0 {k : 0 < k < n, |t2k+1 — a«| < 2^} and
G = {k}J!=0 \ G. Rewrite (22) by dividing it into two sums:
S1 — s. = E д2 f ) + E Д2 f ((2* )•
fceG
fceG
1
c
2
2
a =x
a — x
2
2
2
For every k G G we have |t2k+1 — a«| > 2n/N (0 ^ i ^ m) so the points t2k, t2k+i, t2k+2 are inside some interval (ai5ai+1) and the function f G C^ has an absolutely continuous derivative f'' on (a^, ai+1), therefore we can write |A2f(t2k)| ^ 2maxxG[_n,n] |f''(x)|. For k G G we can write |A2f(t2k)| ^ c(f)/N, also note that |G| ^ 2m. Therefore, we have
|Si| = ^^ < f. (23)
From (9)
2n_1 n_1
ai,2n)(f) = f (tk) cosnk = N ^ (f (i2fc) — f (i2fc+1)) = NS1. (24)
k—0 k—0
From (23) and (24) follows a(N)(f) < c(f )/N2. □
4. THE PROOF OF THEOREM 1
The proof of Theorem 1 consists of two parts: first we prove that the inequalities (3) and (4) of the theorem hold, then we prove that these estimates cannot be improved for all f G C2.
From the inequalities (10), (11), the estimates (13), (14), (20), (21) and Lemma 9 we can easily get (3) and (4). To prove that the order of these estimates cannot be improved we consider the aforementioned 2n-periodic function f (x) = |x|, x G [—n,n]. Obviously, f G . Consider only the case when n < N/2. From (5) follows the inequality |f(x) — Ln,N(f,x)| ^ |f(x) — Sn(f,x)| — |Rn,N(f,x)|. From (20) follows |Rn,N(f, x)| ^ c(f)/N. Therefore, for every e > 0 we can find a natural number N0, such that for every N > N0 follows |Rn,N(f,x)| < e. Let N0(n) be a natural number such that for every N > N0 (n)
max |Rn,N(f,x)| ^ 1 max |f (x) — Sn(f,x)| ,
xGE 2 xGE
N>N0 (n)
where E c R. So, we can write
max |f(x) — Ln,N(f, x)| ^ 1 max |f (x) — Sn(f, x)|. (25)
XGE 2 xGE
N>No(n)
Lemma 10. 7he following inequalities hold:
max |f (x) — Sn(f,x)| ^ c(f)/n, x G R,
xGR
max |f (x) — Sn(f, x)| ^ c(f, e)/n2, |nk — x| ^ e.
Proof. From [14, p. 443] we have the following representation:
, , , n 4 ^ cos(2k — l)x r ,
f(x) = |x| = 2 — n £ (2k — I)* ' x G [—n'n]'
k—1
From the previous equation we can get f (x) - (f,x) = -4 ^м--)^ • For
x = 0 we have
4 1
R(f,0)1 = 4 £ (2k—1)2 > c/n
k=n+1 V J
Now we consider the case when x = n/4 and n + 1 = 41, 1 G N. It is easy to show that
R f q = _2^2 v ( 1 i 1___1___—^ =
ПУ "4) п (8k - 1)2 (8k + 1)2 (8k + 3)2 (8k + 5)V
8k + 1 8k + 3 \
+
n y (8k - 1)2 (8k + 3)2 (8k + 1)2 (8k + 5)2 / '
Hence we have |Rn f, | ^ c/n2. □
From (25) and the above lemma follows
max f (x) - (f, x) | ^ C, x G R,
N>N0 (n)
max |f(x) - Ln,N(f,x)| ^ ^, |x - nk| ^ 6.
N>N0 (n)
Theorem 1 is proved. References
1. Bernshtein S. N. O trigonometricheskom interpolirovanii po sposobu naimen'shih kvadratov [On trigonometric interpolation by the method of least squares]. Dokl. Akad. Nauk USSR, 1934, vol. 4, pp. 1-5 (in Russian).
2. Erdos P. Some theorems and remarks on interpolation. Acta Sci. Math. (Szeged), 1950, vol. 12, pp. 11-17.
3. Kalashnikov M. D. O polinomah nailuchshego (kvadraticheskogo) priblizheniya v zadannoy sisteme tochek [On polynomials of best (quadratic) approximation on a given system of points]. Dokl. Akad. Nauk USSR, 1955, vol. 105, pp. 634-636 (in Russian).
4. Krilov V. I. Shodimost algebraicheskogo interpolirovaniya po kornyam mnogochlena Chebisheva dlya absolutno neprerivnih funkciy i funkciy s ogranichennim izmeneniyem [Convergence of algebraic interpolation with respect to the roots of a Chebyshev polynomial for absolutely continuous functions and functions with bounded variation]. Dokl. Akad. Nauk USSR, 1956, vol. 107, pp. 362-365 (in Russian).
5. Marcinkiewicz J. Quelques remarques sur l'interpolation. Acta Sci. Math. (Szeged), 1936, vol. 8, pp. 127-130 (in French).
6. Marcinkiewicz J. Sur la divergence des polynomes d'interpolation. Acta Sci. Math. (Szeged), 1936, vol. 8, pp. 131-135 (in French).
7. Natanson I. P. On the Convergence of Trigonometrical Interpolation at Equi-Distant Knots. Annals of Mathematics, Second Ser., 1944, vol. 45, no. 3, pp. 457-471. DOI: http://doi.org/10.2307/1969188
8. Nikolski S. M. Sur certaines methodes d'approximation au moyen de sommes trigonome?triques. Izv. Akad. Nauk SSSR, Ser. Mat., 1940, vol. 4, iss. 4, pp. 509-520 (in Russian).
9. Turetskiy A. H. Teorija interpolirovanija v zadachah [Interpolation theory in exercises]. Minsk, Vissheyshaya Shkola Publ., 1968. 320 p. (in Russian).
10. Zygmund A. Trigonometric Series. Vol. 1. Cambridge, Cambridge Univ. Press, 1959. 747 p.
11. Akniyev G. G. Discrete least squares approximation of piecewise-linear functions by trigonometric polynomials. Issues Anal., 2017, vol. 6 (24), iss. 2, pp. 3-24. DOI: http://doi.org/10.15393/j3.art.2017.4070
12. Sharapudinov I. I. On the best approximation and polynomials of the least quadratic deviation. Anal. Math., 1983, vol. 9, iss. 3, pp. 223-234.
13. Sharapudinov I. I. Overlapping transformations for approximation of continuous functions by means of repeated mean Valle Poussin. Daghestan Electronic Mathematical Reports, 2017, iss. 8, pp. 70-92.
14. Courant R. Differential and Integral Calculus. Vol. 1. New Jersey, Wiley-Interscience, 1988. 704 p.
Cite this article as:
Akniyev G. G. Approximation of Continuous 2n-Periodic Piecewise Smooth Functions by Discrete Fourier Sums. Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform., 2019, vol. 19, iss. 1, pp. 4-15. DOI: https://doi.org/10.18500/1816-9791-2019-19-1-4-15
УДК 517.521.2
Приближение непрерывных 2п-периодических кусочно-гладких функций
дискретными суммами Фурье
Г. Г. Акниев
Акниев Гасан Гарунович, младший научный сотрудник, Дагестанский научный центр РАН, 367025, Россия, Махачкала, ул. М. Гаджиева, д. 45, hasan.akniyev@gmail.com
Пусть N > 2 — некоторое натуральное число. Выберем на вещественной оси N равномерно расположенных точек = 2nk/N + u (0 < k < N - 1). Обозначим через (f) = (f, x) (1 < n < N/2) тригонометрический полином порядка n, обладающий наименьшим квадратичным отклонением от f относительно системы ■ Выберем m + 1 точку
-п = a0 < a1 < ... < am-1 < am = п, где m > 2, и обозначим О = {a«}™0. Через C^ обозначим класс 2п-периодических непрерывных функций f, r-раз дифференцируемых на каждом сегменте Д« = [a, ai+1], причем производная f(r) на каждом Д« абсолютно непрерывна. В данной работе рассмотрена задача приближения функций f е полиномами (f, x). Показано, что вместо оценки |f (x) - (f, x)| < c ln n/n, которая следует из известного неравенства Лебега, найдена точная по порядку оценка |f (x) - (f, x)| < c/n (x е R), которая равномерна относительно n (1 < n < N/2). Кроме того, найдена локальная оценка |f (x) - (f, x)| < c(e)/n2 (|x - a«| > e), которая также равномерна относительно n (1 < n < N/2). Доказательства этих оценок основаны на сравнении дискретных и непрерывных конечных сумм ряда Фурье.
Ключевые слова: приближение функций, тригонометрические полиномы, ряд Фурье. Поступила в редакцию: 22.05.2018 / Принята: 28.11.2018 / Опубликована онлайн: 28.02.2019
Образец для цитирования:
Akniyev G. G. Approximation of Continuous 2n-Periodic Piecewise Smooth Functions by Discrete Fourier Sums [Акниев Г. Г. Приближение непрерывных 2п-периодических кусочно-гладких функций дискретными суммами Фурье] // Изв. Сарат. ун-та. Нов. сер. Сер. Математика. Механика. Информатика. 2019. Т. 19, вып. 1. С. 4-15. DOI: https://doi.org/l0.18500/1816-9791-2019-19-1-4-15
